Chapter 7 Sampling Methods

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Chapter 7Sampling Methods The Central Limit
Theorem

• welcome to Inferential Statistics
• (Pulling samples to draw conclusions about a
  population)

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Sampling Distribution of a Sample Mean

• A distribution is a collection of measurements
• A sampling distribution is a collection of
  measurements from a sample
• A sampling distribution of the sample mean is a
  collection of sample means
• How should we describe the sampling distribution
  of the sample mean?
• mean (expected value) stdev (std error)

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Sampling Distribution of a Sample Mean

• Why do we sample?
• GOAL OF SAMPLING
• For the mean of the sampling distribution to be
  equal to the true population mean
• E(X) ?
• if we sample from a population, we can expect the
  behavior of the sampling distribution to be just
  like the behavior of the population
• An unbiased sample

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Now that we have sampled, how can we describe the
population? Use probabilities! Central Limit
Theorem
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Central Limit Theorem

• If samples of a given size are selected from any
  population, the sampling distribution of the
  sample mean will be normally distr.

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Central Limit Theorem

• If samples of a given size are selected from any
  population, the sampling distribution of the
  sample mean will be normally distr.

mean, E(X) ?
standard error, SE s/sqrt(n)
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Central Limit Theorem

• If samples of a given size are selected from any
  population, the sampling distribution of the
  sample mean will be normally distr.
• As n gets larger and larger, this approximation
  gets better
• Wheel of Fortune example

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How determine distribution of outcomes?
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Now that we know the sampling distr. of the
sample mean is normal, what will that allow us to
do?? Calculate probabilities Calculate z-scores
(standardize the data)
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Standardize the Sample Means

• We can calculate probabilities (of the likelihood
  of various sample means) by standardizing
• z-score, z

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Standardize the Sample Means

• We can calculate probabilities (of the likelihood
  of various sample means) by standardizing
• z-score, z

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Standardize the Sample Means

• We can calculate probabilities (of the likelihood
  of various sample means) by standardizing
• need to correct for the mean and stdev of our
  sampling distr., z
• EXAMPLES 7.6, 7.7, 7.8

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Sample Proportions

• The population proportion tells us the proportion
  of the population with a particular attribute,
• where N is the total in the population, and X is
  total in the population with the attribute of
  interest
• The sample proportion tells us the proportion of
  the sample with a particular attribute,
• where n is the total in the sample and x is total
  in the sample with the attribute of interest

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THE POPULATION PROPORTION

• Population proportion
• Sample proportion
• the sample proportion is also normally
  distributed
• E(X) p mean
• std error

EXAMPLES 7.15, 7.17, 7.18
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Standardize the Sample Proportions

• We can calculate probabilities (of the likelihood
  of various sample proportions) by standardizing
  
• need to correct for the mean and stdev of our
  sampling distr., z
• EXAMPLES 7.6, 7.7, 7.8

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Chapter 8

• Estimation Confidence Intervals
• (how do we use the estimates we get from samples?)

Means
Proportions
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• Confidence intervals provide a range in which you
  are sure the true value resides
• you are sure that the population parameter is
  within this interval with a given level of
  confidence
• CONFIDENCE INTERVAL FORMULA

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?
CONFIDENCE LEVELS

• The central limit theorem tells us that the
  sampling distr. of the sample mean is normal
• This means that 95 of sample means will be
  within 2 stdevs of the population mean

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95
99.7
-3 -2 -1 0 1 2 3
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ERROR 1 CONFIDENCE
?

• What error level does a 95 level of confidence
  correspond to?
• ? is the error level
• ? 1 - 0.95
• ? 0.05

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CRITICAL MEASURES come from a distribution
?

• What z-score corresponds to an error level?
• consider that the error is on BOTH sides of the
  mean (half of the error is on one side, and half
  is on the other side of the mean)

95
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CRITICAL MEASURES come from a distribution
?

• What z-score corresponds to an error level?
• consider that the error is on BOTH sides of the
  mean
• We need to find a z-score that corresponds to
  half of the error (?/2)
• Using EXCEL, what z-value corresponds to ?/2?
• Use the NORMSINV function

EXAMPLES find zCRIT for p 90, 99
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STANDARD ERROR comes from the measurements
?
?
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CONFIDENCE INTERVALS FOR THE POPULATION
MEAN(stdev known OR larger samples ngt120)

• The point estimate is the sample mean
• The critical value is the z-score or zCRIT
• Need to know the confidence level, p (? 1 p)
• NORMSINV(1-?/2) in Excel
• The standard error, SE ?/sqrt(n)

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CONFIDENCE INTERVALS FOR THE POPULATION
MEAN(stdev known OR larger samples ngt120)

• EXAMPLES 8.1, 8.7

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GENERAL FORMULA FOR CONFIDENCE INTERVALS
The std error is the std dev of the sampling
distribution
A summary measure from a sample
A critical measure related to the desired level
of confidence
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CONFIDENCE INTERVALS FOR THE POPULATION MEAN
(stdev unknown AND smaller samples nlt120)

• The point estimate is the sample mean
• The measure of confidence or critical value is
  the z-score or zCRIT
• We dont know the stdev, so we must use our
  approximation to the population stdev
• The z-score is now represented as
• With this estimation (substitution), the sampling
  distr. is no longer normal!!

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CONFIDENCE INTERVALS FOR THE POPULATION
MEAN(stdev unknown AND smaller samples nlt120)

• The point estimate is the sample mean
• The measure of confidence or critical value is
  the z-score or zCRIT
• We dont know the stdev, so we must use our
  approximation to the population stdev
• The z-score is now represented as
• With this estimation (substitution), the sampling
  distr. is no longer normal!!

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t-DISTRIBUTION

• The t-distribution has one parameter, degrees of
  freedom
• df of independent obs. - of estimated
  parameters
• df n - 1
• The t-distr. is symmetric and bell-shaped, but
  has a larger area in the tails (than the normal
  distr.)
• when n (sample size) is very large, the t-distr.
  approaches the normal distr.

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CONFIDENCE INTERVALS FOR THE POPULATION
MEAN(stdev unknown AND smaller samples nlt120)

• The point estimate is the sample mean
• The measure of confidence or critical value is
  the t-score or tCRIT
• Need to know a confidence level, p (? 1 p)
• TINV(?, df) in Excel

EXAMPLE 8.10ac
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CONFIDENCE INTERVALS FOR THE POPULATION
MEAN(stdev unknown AND smaller samples nlt120)

• The point estimate is the sample mean
• The measure of confidence or critical value is
  the t-score or tCRIT
• Need to know a confidence level, p (? 1 p)
• TINV(?, df) in Excel
• The standard error, SE s/sqrt(n)

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CONFIDENCE INTERVALS FOR THE POPULATION
MEAN(stdev unknown AND smaller samples nlt120)

• EXAMPLES 8.15, 8.16, 8.22

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GENERAL FORMULA FOR CONFIDENCE INTERVALS
The std error is the std dev of the sampling
distribution
A summary measure from a sample
A critical measure related to the desired level
of confidence
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CONFIDENCE INTERVALS FOR THE POPULATION PROPORTION

• The point estimate is the sample proportion
• The measure of confidence or critical value is
  the z-score or zCRIT
• Need to know a confidence level, p (? 1 p)
• NORMSINV(1-?/2) in Excel

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CONFIDENCE INTERVALS FOR THE POPULATION PROPORTION

• EXAMPLES 8.28 , 8.30